Real functions Functions of several real variables Theorem

Inverse function theorem

Builds on C^k function characterised by continuous derivatives Jacobian determinant Fréchet derivative Derivative of the inverse function

Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $k \ge 1$ , and let $f \colon U \to \mathbb{R}^n$ be of class C^k on $U$ . Let $a \in U$ , and suppose that the derivative $Df(a)$ is invertible, that is, the Jacobian determinant of $f$ at $a$ is not zero. Then there are open sets $V \subseteq U$ with $a \in V$ and $W \subseteq \mathbb{R}^n$ with $f(a) \in W$ such that $f$ restricted to $V$ is a bijection from $V$ onto $W$ . Its inverse $f^{-1} \colon W \to V$ is of class $C^k$ , and $D(f^{-1})(f(x)) = Df(x)^{-1}$ for each $x \in V$ .

Remarks

The theorem is local. The map $f(x, y) = (e^x \cos y, e^x \sin y)$ has invertible derivative at each point of $\mathbb{R}^2$ , but it is not injective, since $f(x, y + 2\pi) = f(x, y)$ . Continuity of the derivative matters. The function $g(x) = x + 2 x^2 \sin(1/x)$ , $g(0) = 0$ , has $g'(0) = 1$ , but $g$ is not injective on any interval around $0$ . A standard proof applies the contraction mapping principle to $x \mapsto x + Df(a)^{-1}(y - f(x))$ near $a$ . The formula for the derivative of the inverse is the derivative of the inverse function.

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Euclidean space object
Function concept
Function composition concept
Fréchet derivative definition
Function of several variables concept
Limit of a function of several variables concept
Partial derivative concept
Differentiable function of several variables definition
Partial derivative definition
Differentiable on an open set definition
Jacobian matrix definition
Chain rule in several variables statement
Higher partial derivatives definition
Iterated derivative definition
Jacobian determinant definition
Derivative of the inverse function statement
C^k function characterised by continuous derivatives statement
Inverse function theorem statement

Where this leads

38 pages build on this one.

Tag	`0K9HSJFCMJ`
URI	`m2://0K9HSJFCMJ`
Aliases	`inverse-mapping-theorem`
Kind	statement, theorem
Writing	W1 stated
MSC	26B10
Status	This copy of the site has no build data. Run `m2 status 0K9HSJFCMJ`.
Uses	C^k function characterised by continuous derivatives, Jacobian determinant, Fréchet derivative, Derivative of the inverse function
Same as	Wikidata Q931001 Wikipedia Inverse function theorem

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